Partial sum of bounded series is Cauchy in $C(X, \mathbb{R})$?

Question

I am reading a proof of Tietze's Extension Theorem and there was a claim that, given a sequence of functions $h_n(x) : X \to \mathbb{R}$

If $$G = \sum\limits_{n = 1}^\infty h_n(x)$$

Is bounded, then the partial sums of $G$ are Cauchy sequences on $C(X, \mathbb{R})$.

I have never seen this before, is it saying that $\{h_1, h_1+h_2, h_1+h_2+h_3,\ldots\}$ is Cauchy? And why?

Thanks in advance

Answer 1

This is false as stated. Counterexample: On $[0,1],$ let $h_n(x) = \sin(2\pi/x)\cdot\chi_{[1/(n+1),1/n]}.$ Then $\sum_{n=1}^{\infty} h_n(x) = \sin(2\pi/x)\cdot \chi_{(0,1]}.$ If $S_n$ is the $n$th partial sum of this series, then $\|S_{n+1} - S_n\|_\infty = 1$ for all $n,$ hence $S_n$ is not Cauchy in $C([0,1]).$